Brad Larsen wrote:
On Wed, 19 Dec 2007 02:00:53 -0500, Jules Bean <[EMAIL PROTECTED]>
wrote:
Brad Larsen wrote:
Hi there list,
How would one go about creating a new type for a subset of the
integers, for (contrived) example just the even integers? I was
thinking of making a new type
newtype EvenInt = EvenInt Integer
but the problem with this is that it accepts any integer, even odd
ones. So to prevent this, the module does not export the constructor
for it---rather, the module exports a function `mkEvenInt' that
creates an EvenInt if the given value is acceptable or raises an
error otherwise.
What's the right way to do this? Thanks!
There are two ways:
(1) Have a representation which admits invalid values, and provide
combinators which only perfect validity, and prove that consumers
using your combinators can't produce invalid values.
(2) Have a cleverly designed representation such that every
representation is valid.
An example here, for (2) would be to store n/2; there is a bijection
between 'Integer' and 'EvenInt' given by n/2.
To make sure I understand, an example of (1) would be to export only a
``smart constructor'' that somehow converts invalid values to valid ones
(say, add 1 to arguments that are odd)?
I just meant one that throws a runtime error if they are not.
In your example of 2, how would you go about storing n/2? Store just n
as in (newtype EvenInt = EvenInt Integer) and then write all functions
that deal with EvenInts so that they account for the division by two?
You would store n/2. So "8" would be represented as EvenInt 4, "14" as
EvenInt 7. Then you write the num instance to account for the division.
Check out the num instance in Data.Fixed for an example.
In real, more complex problems, (2) often isn't possible and we resort
to (1). E.g. the representation of balanced trees (AVL? RedBlack?)
admits invalid values (both unbalanced trees and out-of-order trees)
and we rely on the reduced set of combinators never to generate one.
Jules
In my particular case, or what I actually want to do, is define a finite
segment of the integers (0-42, say) as a new type and have that checked
at compile time. Any way of doing this w/o defining Peano numbers or a
whole bunch of nullary constructors (i.e. I'm hoping to be able to
define a type whose constructor will accept only Integer arguments
between 0 and 42)?
Not at compile-time, no. The best you can do is runtime error.
There is no compile-time facility to check that an Int lies in a
particular range. Consider, for example, that the int might not be known
at compiletime:
do
id <- myThreadID
return (MkSmallInt id)
^^ how can the compiler know?
In general, of course, the question of calculating if an int is under 42
at compile time is the question of partial evaluation. Partial
evaluation at compile time is flat-out impossible for IO actions, and
although it's possible for pure functions, GHC doesn't do it to any
significant extent. (It would make the compiler potentially
non-terminating, of course!)
Jules
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